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In a quadratic function, the vertex of a parabola plays a crucial role because it gives a precise point where the function reaches its lowest or highest value. For a function in the form of \( y = ax^2 + bx + c \), the vertex can be found using the formula \( x = \frac{-b}{2a} \). This formula calculates the \( x \)-coordinate of the vertex by averaging the roots of the equation when it is set to zero.
If you notice, the vertex position’s formula derives from symmetry. Since a parabola is symmetric around a vertical axis, the axis of symmetry essentially splits it into mirror images. Calculating \( x = \frac{-b}{2a} \) helps you find this axis.
Moreover, the sign of \( a \) is significant here. If \( a > 0 \), the parabola opens upwards, and the vertex represents the minimum value. Conversely, if \( a < 0 \), the parabola opens downwards, and the vertex shows the maximum value. Understanding and identifying the vertex is a game-changer for analyzing quadratic functions and solving real-world problems.

When solving quadratic equations, the quadratic formula is indispensable. The general quadratic equation is expressed as \( ax^2 + bx + c = 0 \). The quadratic formula provides the solutions, or roots, of the equation through:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Plugging the values of \( a \), \( b \), and \( c \) from your equation helps find the precise points where the quadratic touches or crosses the x-axis. It is a reliable method, as it applies to any quadratic equation regardless of its reducibility by factoring.
Understanding the underlying discriminant part \( b^2 - 4ac \) is also essential with the quadratic formula. The value determines the nature of the roots:

• If it is positive, there are two distinct real roots.
• Zero results in exactly one real root.
• Negative means no real roots, only complex ones.

Applying the quadratic formula, you can uncover the comprehensive behavior of the parabolic curve which aids in graphing and understanding quadratics effectively.

Quadratic functions inherently possess either minimum or maximum values that occur at their vertex. The value depends on the form and direction of the parabola described by the equation.
For the function \( g(x) = 100x^2 - 1500x \) given in the exercise, \( a = 100 \) is greater than zero. Therefore, the parabola opens upwards, which means the function has a minimum value rather than a maximum. The vertex \( (x, g(x)) \) captures this peak of the parabola. By determining \( x \) from \( x = \frac{-b}{2a} \), we find that \( g(x) \) has a minimum at \( x = 7.5 \).
Plugging this \( x \)-coordinate back into the original function delivers the minimum value, which calculates to \( -5625 \).
This emphasizes not only the crucial role vertices and quadratic characteristics play in defining minima and maxima but also highlights their practical application in real-world situations like finding optimal values in economics or physics.